Numerical study of the free energy monotonicity for the Enskog equation: The thermal relaxation of the dense gas between two parallel plates

arXiv:2603.23839 · cond-mat.mes-hall · Submitted 2026-03-25 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Numerical study of the free energy monotonicity for the Enskog equation".

Mira: Here's the gist of the paper. The gist The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog equation.

Kai: Model and Equations The study investigates a dense gas in a gap between two parallel plates maintained at a common uniform constant temperature T0 The gas behavior is described by the Enskog equation.

Mira: Two types of Enskog equation are used, one is the original Enskog equation with the original Enskog factor, while the other is that with a modified Enskog factor proposed recently in Takata & Takahashi, Phys.

Kai: Rev.

Title and authors: Kai: So we're diving into "Numerical study of the free energy monotonicity for the Enskog equation: The thermal relaxation of the dense gas between two parallel plates," and who wrote this work.

Mira: It’s by Takata, Sakata, and Takahashi from Kyoto University; they are exploring how a dense gas relaxes thermally when confined between plates at a constant temperature.

Lev: I'm curious what they built to simulate this—does it involve high-fidelity molecular dynamics or something more tailored for these kinetic equations?

Kai: They’ve used a hybrid numerical method: the finite-difference method for handling derivatives and the fast Fourier spectral method specifically for calculating that collision integral term.

Mira: The core idea they are testing is whether using that modified Enskog factor—the EESM—gives us thermodynamic consistency when modeling these non-equilibrium gas states.

Kai: They set up this problem in dimensionless units, defining Kn as the ratio of the reference mean free path to the system length L, which is a standard way to normalize these kinds of problems.

Lev: When you get into real hardware simulation, those scaling parameters like that Kn and your initial conditions become crucial for determining if your numerical setup even makes physical sense.

Mira: The paper focuses on showing that the free energy decreases monotonically in time when using the EESM version, which is a big deal because it connects their numerical results to established thermodynamic laws.

The paper's summary: Kai: So what they summarize is that they investigated the thermal relaxation problem between parallel plates at a common temperature and looked at how the free energy evolves using both original and modified Enskog equations.

Mira: They found that for the modified factor version, the evaluated free energy monotonically decreases in time, which is different from the original equation where it doesn't necessarily decrease monotonically.

Lev: So they're suggesting that this modification provides a better framework for describing relaxation dynamics in these confined systems compared to what’s currently available.

Kai: They also noted that while the density and temperature profiles at the final steady state are similar between the two versions, there are notable differences in how those profiles evolve during the transient process.

Mira: Specifically, they see clear differences between OEE and EESM in density profiles from timeˆ equals zero point zero one to zero point one, and also a difference in temperature evolution at timeˆ equals zero point zero eight for the paper "Numerical study of the free energy monotonicity for the Enskog equation: The thermal relaxation of the dense gas between two parallel plates."

The paper's improvements: Kai: They suggest that their main improvement is using the Enskog equation with a slight modification, which they call EESM, because it newly provides thermodynamic consistency to the numerical studies based on the Enskog equation.

Mira: That modification addresses a fundamental issue: it allows them to ensure that the H-theorem holds for this system, which was missing in their earlier work using the original OEE.

Lev: If we were trying to run this on actual error-corrected hardware, having that thermodynamic consistency built into the equation means you wouldn't have to worry as much about artifacts arising from non-physical relaxation paths.

Kai: They also highlight that the non-kinetic contribution is what’s crucial for the monotonicity in time evolution of EESM, which points toward how those collisional effects are behaving in this dense environment.

Mira: So they're pointing out that the structure of the modified factor is important because it introduces a term that keeps things physically consistent during the relaxation process.

Conclusion: Kai: To wrap up, they provide numerical evidence supporting the theoretical conclusion that free energy decreases monotonically in time for EESM, and not necessarily for OEE, while also noting differences in the density profiles during the transient phase.

Mira: The paper concludes that this work provides evidence for the theoretical conclusion on monotonic decrease of free energy mentioned earlier, even though those density profile differences are minor at the final steady state.

Lev: For us, it means if we’re designing systems where thermal relaxation matters, like in micro-scale confinement, we need to use models like EESM to predict stable relaxation paths reliably.

Kai: So "Numerical study of the free energy monotonicity for the Enskog equation: The thermal relaxation of the dense gas between two parallel plates" shows that a small modification to the collision factor can make a big difference in predicting system behavior over time.

Mira: Yeah, it’s about how we extend thermodynamic free energy into non-equilibrium states and find better ways to describe those dynamics for dense gases.

Lev: It's interesting because it shows that even with complex kinetic equations, if you add the right term, you can guarantee a physical direction for relaxation.

Kyoto University

cond-mat.mes-hall

Submitted: 2026-03-25

Updated: 2026-10-21

Comments: 21 pases, 8 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog

Key concepts

Enskog Equation: This is a mathematical model used to describe how dense, hard-sphere gas molecules behave when they are confined between two parallel plates. It accounts for the effects of molecular collisions, which are important in dense systems like gases in small gaps.
Free Energy Monotonicity: This refers to whether the total free energy of the system consistently decreases as time passes during its relaxation process. The study shows that this decrease is guaranteed only when using a specific modified version of the Enskog equation, highlighting a key difference between theoretical models.
H-theorem: The H-theorem is a fundamental principle in statistical mechanics that dictates the direction of irreversible processes, specifically stating that entropy (or related quantities like free energy) must increase or decrease over time. The paper tests whether this theorem holds true for different versions of the Enskog equation.

Terminology

Monotonicity — 1× in this paper · explained in 2 episodes

Summary

The gist The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog equation.

Model and Equations

The study investigates a dense gas in a gap between two parallel plates maintained at a common uniform constant temperature T0 The gas behavior is described by the Enskog equation. Two types of Enskog equation are used, one is the original Enskog equation with the original Enskog factor, while the other is that with a modified Enskog factor proposed recently in Takata & Takahashi, Phys. Rev. E 111, 065108 (2025). The gas molecules are hard-spheres with a common diameter σ and mass m. The one-particle velocity distribution function (VDF) of gas molecules is denoted by f(t, X1, ξ). The Enskog equation is written as ∂f/∂t + ξ1∂f/∂X1 = J(f).

Key Theoretical Concepts

The evaluation of the free energy is a natural extension of the thermodynamic free energy to the non-equilibrium state. It is observed that this free energy monotonically decreases in time for the modified factor version, while it is not necessarily the case for the original version. The H-theorem was shown to hold not for OEE but for the modified (or revised) Enskog equation [15, 16]. The Enskog equation with a slight modification (EESM) can newly provide thermodynamic consistency to the numerical studies based on the Enskog equation.

Numerical Methodology

The numerical method is a hybrid of the finite-difference method for derivatives and the fast Fourier spectral method for collision integral. The problem is analyzed in dimensionless nomenclature, where Kn is defined as Kn =l0/L with l0 being the following reference mean free path. The initial- and boundary-value problem (14)–(16) is thus characterized by four dimensionless parameters ˆσ ≡ σ/L, η0 ≡ πσ3ρ0/(6m), λ, and w. The fast Fourier spectral method is adopted for the computations of collision integral. A process of controlling total mass is preferred in the present study by a simple normalization of the total mass at the end of every time step.

Results on Free Energy Monotonicity

Figure 3 shows the time evolution of F in the case of η0 = 0.25, σ/L = 0.1 (Kn = 0.0227), and λ = 0.1 for different values of w. It is observed that F is almost identical between EESM and OEE when w = 0.1. However, as w increases, they gradually deviate from each other around between tˆ = 2 × 10−3 and 4 × 10−2. Moreover, F decreases monotonically in the case of EESM, but not monotonically in the case of OEE. This is a clear numerical evidence for that the H-theorem in [21] does not hold actually in the latter case. The non-kinetic contribution is crucial for the monotonicity in the time evolution of EESM.

Density and Temperature Profiles

The density and temperature profiles at the final steady state do not depend on the parameters λ and w in the initial condition. At the steady state, the temperature is uniform and is the same as the plate temperature. However, the density is not uniform. The difference between OEE and EESM is minor for density and temperature profiles at the final steady state but a notable difference for F exists. During the transient process, the density profiles are clearly different between EESM and OEE, in particular from tˆ = 0.01 to tˆ = 0.1. The time evolution of temperature also shows a difference between EESM and OEE at tˆ= 0.08.

Conclusion

The present numerical results provide the evidence that supports the theoretical conclusion on the monotonic decrease of the free energy in [21]. The differences of the density profiles are also observed during the time evolution, although the profiles at the final steady states are close to each other. The present work is concluded by observing that F decreases monotonically in time for EESM and not necessarily for OEE.

ACKNOWLEDGMENTS

The present work is supported in part by the Kyoto University Foundation, by the JSPS Grant-in-Aid (No. 24KJ1450), and by the HPCI system Research Project (No. hp250004). It is partly achieved by the use of SQUID at D3 Center, The University of Osaka.

Appendix A: Dimensionless nomenclature

The dimensionless initial condition (16) and the boundary condition (15) are obtained straightforwardly from the definitions of tˆ, x, ζ, and ˆf. It is clear that the details of gˆ do not induce parameters other than ˆσ and η0. Hence, even by taking account of the two parameters w and λ in the initial condition, there are only four independent parameters in the present problem regardless of whether it is OEE or EESM.

REFERENCES

[1] C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988

[2] Y. Sone, Molecular Gas Dynamics, Birkh¨auer, Boston, 2007

[3] D. Enskog, Kinetic theory of heat conduction, viscosity and self-diffusion in compressed gases and liquids, in Kinetic Theory, Vol. 3, S. G. Brush ed., Pergamon Press, Oxford, Part 2

[4] R. Soto, Kinetic Theory and Transport Phenomena

[5] A. Frezzotti, Molecular dynamics and Enskog theory calculation of one dimensional problems in the dynamics of dense gases, Physica A 240, 202–211 (1997)

[6] A. Frezzotti, Monte Carlo simulation of the heat flow in a dense hard sphere gas, Eur. J. Mech. B/Fluids 18, 103–119 (1999)

[7] M. Kon, K. Kobayashi and M. Watanabe, Method of determining kinetic boundary conditions in net evaporation/condensation, Phys. Fluids 26, 072003 (2014)

[8] L. Wu, H. Liu, J. M. Reese and Y. Zhang, Non-equilibrium dynamics of dense gas under tight confinement, J. Fluid Mech. 794, 252–266 (2016)

[9] A Frezzotti, L Gibelli, D A Lockerby and J E Sprittles, Mean-field kinetic theory approach to evaporation of a binary liquid into vacuum, Phys. Rev. Fluids 3, 054001 (2018)

[10] M Hattori, S Tanaka and S Takata, Heat transfer in a dense gas between two parallel plates, AIP Advances 12, 055220 (2022)

[11] J. M Montanero and A Santos, Monte Carlo simulation method for the Enskog equation, Phys. Rev. E 54, 438–444 (1996)

[12] A Frezzotti, A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids 9, 1329–1335 (1997)

[13] J. M Montanero and A Santos, Simulation of the Enskog equation a la Bird, Phys. Fluids 9, 2057–2060 (1997)

[14] L Wu, Y Zhang and J M Reese, Fast spectral solution of the generalized Enskog equation for dense gases, J. Comput. Phys. 303, 66–79 (2015)

[15] H van Beijeren and M H Ernst, The modified Enskog equation, Physica 68, 437–456 (1973)

[16] P Resibois, H-theorem for the (modified) nonlinear Enskog equation, J. Stat. Phys. 19, 593–609 (1978)

[17] P Maynar, M I Garcia de Soria and J J Brey, The Enskog equation for confined elastic hard spheres, J. Stat. Phys.

Improvements for AI systems

Improved simulation of non-equilibrium gas flow by accurately modeling free energy monotonicity in dense gases using a modified Enskog factor (EESM). This allows AI systems to predict whether a system will naturally relax to equilibrium or exhibit transient non-monotonic behavior under specific conditions.
Development of density and temperature profile prediction for confined, dense gases by employing the Enskog equation with the EESM variant. This enables AI models to generate realistic spatial distributions for gas density and temperature within microfabrication gaps, even when initial conditions vary significantly.
Enhanced kinetic theory modeling capabilities by integrating the H-theorem validation into numerical solvers for the modified Enskog equation. This allows AI systems to rigorously verify thermodynamic consistency in non-equilibrium simulations, ensuring that predicted relaxation paths adhere to established physical laws.

Abstract

The thermal relaxation problem between two parallel plates with the same temperature is investigated, aiming to study the behavior of the free energy of the dense gas described by the Enskog equation. Two types of Enskog equation have been used: one is the Enskog equation with the original Enskog factor, while the other is that with a modified Enskog factor proposed recently in Takata & Takahashi, Phys. Rev. E 111, 065108 (2025). The evaluated free energy is a natural extension of the thermodynamic free energy to the non-equilibrium state. It is observed that this free energy monotonically decreases in time for the modified factor version, while it is not necessarily the case for the original version. Differences are also observed in other quantities in the time evolution, most typically in the density profile.